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Subsections

• Some definitions
• Labels, sub- and super-scripts
• Coordinate frames
• Matrices and vectors
• Miscellaneous parameters

APPENDIX: CONVENTIONS

A consistent nomenclature and precise definitions are extremely important for a software package like AIPS++, which aspires to be a `world reduction package', and to which workers with a large spacetime separation are supposed to contribute. One of the most sensitive areas in this respect is the Measurement Equation, which underlies the central subject of uv-calibration and imaging.

However, it is not easy to define, adopt and enforce the use of a suitable set of conventions. This appendix is a hopefully useful step in that process. It proposes coordinate conventions and some definitions (notably the one for feed!), and lists symbols that have been defined in a separate TeX file (referred to as \include(megi-symbols) in this LaTeX document). The TeX syntax is shown in small print (e.g. \FeedI), for easy reference.

Some definitions

The following definitions are displayed in a distinctive font throughout the text of this document in order to emphasize that they have been defined explicitly.

• A receptor (\Receptor) converts the incident electric field into a voltage.

• An IF-channel (\IFchannel) is one of the two output signals of a feed, one for each `polarisation'. NB: The signals in a pair of IF-channels may be a linear combination of the signals of the two receptors.

• A feed (\Feed) is the most fundamental concept of the M.E. formalism, since Jones-matrices are feed-based. Although a feed may sometimes have only one receptor, it usually has two, which is necessary and sufficient to fully sample the incident e.m. field. Each feed is modelled by its own Jones matrix. NB: A feed is a logical concept. Thus, the same physical feed may be involved in several logical feeds, e.g. for different beams in a multi-beam instrument, or for different spectral windows.

• An antenna (\Antenna) is a physical grouping of feeds. NB: As a concept, it tends to play a rather confusing role in the M.E. discussions.

• An interferometer (\Interferometer) is the combination of two feeds. Its output is a visibility of 1-4 spectra, depending on the number of IF-channels per feed. NB: Sometimes the combination of two individual IF-channels is also called an interferometer. In that case, its output is a single spectrum.

• A telescope (\Telescope) is an entire instrument. It can be a single dish (e.g. GBT) or an aperture synthesis array (e.g. ATCA).

• A projected (\Projected) angle is an angle projected on the plane perpendicular to the propagation direction (the $\sf z$-axis).

Labels, sub- and super-scripts

  

 
$\sf i$,$\sf j$
\FeedI,\FeedJ 		 feed labels 

 
$\sf a$,$\sf b$
\RcpA,\RcpB 		 receptor labels, two per feed. 

 
$\sf p$,$\sf q$
\IFP,\IFQ 		 IF-channel labels,two per feed. 

 
$\sf r$,$\sf l$
\RPol,\LPol 		 circular polarisation (right, left) 

 
$\sf x$,$\sf y$
\XPol,\YPol 		 linear polarisation (N-S, E-W) 

 

 
A+, A$\odot$ A\ssLin,A\ssCir 		 superscripts for linear and circular polarisation 

 
A$\sf i$, A$\sf i$$\sf j$ A\ssI,A\ssIJ 		 feed subscripts

The subscript convention of matrix elements is as follows: $\sf Y_{{\sf i}{\sf b}{\sf p}}^{}$ refers to a matrix element of matrix $\sf Y$ for feed $\sf i$, which models the coupling of the signal going from receptor $\sf b$ to IF-channel $\sf p$.

Coordinate frames

Fig 1 gives an overview of the coordinate system(s) used. All angles on the Sky are measured counter-clockwise, i.e. in the direction North through East. When relevant, `axis' means `positive axis' (e.g. the positive $\sf x$-axis). It is important to make a distinction between:

The beam frame(s): In order to calculate the effects of the primary beam on the signal of a source in direction $\vec{\rho}\,$($\sf l$,$\sf m$), the shape and position of the voltage beams of each receptor on the Sky has to be calculated. For fully steerable parabolic antennas, which have constant beamshapes, this can be done most conveniently in coordinate frames defined by the projected position angles of the receptors. To allow for the fact that the two beams of a feed are closely coupled, an intermediate feed-frame is defined also.

The electrical frame: For the polarisation of the signal, the only relevant parameters are the projected angles w.r.t. the `electrical' axes $\sf x$ and $\sf y$ defined by the IAU.

NB: In order to see that two frames are needed, consider that Faraday rotation rotates the electric vector, but not the beam on the sky.

  


Frame of the entire telescope (single dish or array): 

 $\vec{r}\,$
\vvAntPos 		 Projected feed (receptor?) position vector  

 
$\sf u$,$\sf v$,$\sf w$
\ccU,\ccV,\ccW 		 Projected baseline coordinates  

 $\vec{u}\,$
\vvUVW 		 Projected baseline vector  
$\vec{u}\,$($\sf u$,$\sf v$,$\sf w$)  

 


Electrical frame on the sky (IAU definition): 

 
$\sf x$,$\sf y$
\ccX,\ccY 		 IAU electrical frame on the sky.  

 $\sf z$
\ccZ 		 propagation direction of incident field.  

 
$\gamma_{{\sf x}{\sf y}}$
\aaXY 		 Angle from $\sf x$-axis to $\sf y$-axis (= $\pi$/2) 

 
$\sf x$,$\sf y$
\ccXPol,\ccYPol 		 linear polarisation coordinates.  

 
$\sf r$,$\sf l$
\ccRPol,\ccLPol 		 circular polarisation coordinates.  

 


Sky frame (w.r.t. fringe stopping centre): 

 
$\sf l$,$\sf m$,$\sf n$
\ccL,\ccM,\ccN 		 Coordinates (direction cosines) 

 
$\vec{\rho}\,$
\vvLMN 		 Source direction vector  
$\vec{\rho}\,$($\sf l$,$\sf m$)  

 
$\vec{\rho}_{ftc}$
\vvFTC 		 Fringe Tracking Centre 
$\vec{\rho}_{ftc}$(RA, DEC,$\sf f$)  

 
$\vec{\rho}_{mc}$
\vvMC 		 Map Centre 
$\vec{\rho}_{ftc}$($\sf l$,$\sf m$)  

 
$\gamma_{{\sf l}{\sf m}}$
\aaLM 		 Angle from $\sf l$-axis to $\sf m$-axis (= $\pi$/2) 

 
$\gamma_{{\sf l}{\sf x}}$
\aaLX 		 Angle from $\sf l$-axis to $\sf x$-axis (= $\pi$/2) 

 


Coordinate frame of feed $\sf i$, projected on the sky: 

 
$\sf l^{'}_{{\sf i}}$,$\sf m^{'}_{{\sf i}}$
\ccLI,\ccMI 		 Coordinates 

 
$\sf l_{{\sf i}{0}}^{}$,$\sf m_{{\sf i}{0}}^{}$
\ccLIO,\ccMIO 		 Origin (
$\sf l$,$\sf m$) of feed-frame. 

 
$\gamma_{{\sf l}{\sf i}}$
\aaLI 		 Angle from $\sf l$-axis to 
$\sf l^{'}_{{\sf i}}$-axis 

 
$\gamma_{{\sf x}{\sf i}}$
\aaXI 		 Angle from $\sf x$-axis to 
$\sf l^{'}_{{\sf i}}$-axis (
= -$\gamma_{{\sf l}{\sf x}}$+$\gamma_{{\sf l}{\sf i}}$) 

 


Coordinate frame of receptor $\sf a$
of feed $\sf i$,     projected on the sky: 

 
$\sf l^{''}_{{{\sf i}{\sf a}}}$,$\sf m^{''}_{{{\sf i}{\sf a}}}$
\ccLIA,\ccMIA 		 Coordinates 

 
$\sf l^{'}_{{\sf i}}_{{\sf a}{0}}$,$\sf m^{'}_{{\sf i}}_{{\sf a}{0}}$
\ccLIAO,\ccMIAO 		 Origin (
$\sf l^{'}_{{\sf i}}$,$\sf m^{'}_{{\sf i}}$) of receptor-frame. 

 
$\gamma_{{\sf i}{\sf a}}$
\aaIA 		 Angle from 
$\sf l^{'}_{{\sf i}}$-axis to 
$\sf l^{''}_{{{\sf i}{\sf a}}}$-axis 

 
$\gamma_{{\sf x}{\sf a}}$
\aaXA 		 Angle from $\sf x$-axis to 
$\sf l^{''}_{{{\sf i}{\sf a}}}$-axis (
= -$\gamma_{{\sf l}{\sf x}}$+$\gamma_{{\sf l}{\sf i}}$+$\gamma_{{\sf i}{\sf a}}$) 

 


Coordinate frame of receptor $\sf b$
of feed $\sf i$,     projected on the sky: 

 
$\sf l^{''}_{{{\sf i}{\sf b}}}$,$\sf m^{''}_{{{\sf i}{\sf b}}}$
\ccLIB,\ccMIB 		 Coordinates  

 
$\sf l^{'}_{{\sf i}}_{{\sf b}{0}}$,$\sf m^{'}_{{\sf i}}_{{\sf b}{0}}$
\ccLIBO,\ccMIBO 		 Origin (
$\sf l^{'}_{{\sf i}}$,$\sf m^{'}_{{\sf i}}$) of receptor-frame. 

 
$\gamma_{{\sf i}{\sf b}}$
\aaIB 		 Angle from 
$\sf l^{'}_{{\sf i}}$-axis to 
$\sf l^{''}_{{{\sf i}{\sf b}}}$-axis 

 
$\gamma_{{\sf y}{\sf b}}$
\aaYB 		 Angle from $\sf y$-axis (!) to 
$\sf l^{''}_{{{\sf i}{\sf b}}}$-axis (
= -$\gamma_{{\sf x}{\sf y}}$-$\gamma_{{\sf l}{\sf x}}$+$\gamma_{{\sf l}{\sf i}}$+$\gamma_{{\sf i}{\sf b}}$)

Figure 1: Label: fig-coord  File: fig-coord.eps

The coordinates $\sf l^{''}_{{{\sf i}{\sf a}}}$,$\sf m^{''}_{{{\sf i}{\sf a}}}$ and $\sf l^{''}_{{{\sf i}{\sf b}}}$,$\sf m^{''}_{{{\sf i}{\sf b}}}$ of the frames of receptors $\sf a$ and $\sf b$ in equ 37 are related to the celestial coordinate frame $\sf l$,$\sf m$ in a two-step process. First we define an intermediate feed-frame $\sf l^{'}_{{\sf i}}$,$\sf m^{'}_{{\sf i}}$ for feed $\sf i$, projected on the Sky:

$$\left(\vphantom{\begin{array}{c} {{\sf l}^{'}_{{\sf i}}}\\ {{\sf m}^{'}_{{\sf i}}}\\ \end{array}}\right. \begin{array}{c} {{\sf l}^{'}_{{\sf i}}}\\ {{\sf m}^{'}_{{\sf i}}}\\ \end{array} \left.\vphantom{\begin{array}{c} {{\sf l}^{'}_{{\sf i}}}\\ {{\sf m}^{'}_{{\sf i}}}\\ \end{array}}\right) \sf Rot \gamma_{{\sf l}{\sf i}} \left(\vphantom{\begin{array}{c} {\sf l}- {{\sf l}_{{\sf i}{0}}}\\ {\sf m}- {{\sf m}_{{\sf i}{0}}}\\ \end{array}}\right. \begin{array}{c} {\sf l}- {{\sf l}_{{\sf i}{0}}}\\ {\sf m}- {{\sf m}_{{\sf i}{0}}}\\ \end{array} \left.\vphantom{\begin{array}{c} {\sf l}- {{\sf l}_{{\sf i}{0}}}\\ {\sf m}- {{\sf m}_{{\sf i}{0}}}\\ \end{array}}\right)$$ (54)

in which ($\sf l_{{\sf i}{0}}^{}$,$\sf m_{{\sf i}{0}}^{}$) is the Pointing Centre of feed $\sf i$, and $\sf Rot$($\gamma_{{\sf l}{\sf i}}$) is a rotation over the projected angle $\gamma_{{\sf l}{\sf i}}$ between the positive $\sf l$-axis of the Sky frame and the $\sf l^{'}_{{\sf i}}$-axis of the feed-frame.

The voltage beams themselves are best modelled in a receptor-frame (see equ 37), again projected on the Sky. For receptor $\sf a$ we have:

$$\left(\vphantom{\begin{array}{c} {{\sf l}^{''}_{{{\sf i}{\sf a}}}}\\ {{\sf m}^{''}_{{{\sf i}{\sf a}}}}\\ \end{array}}\right. \begin{array}{c} {{\sf l}^{''}_{{{\sf i}{\sf a}}}}\\ {{\sf m}^{''}_{{{\sf i}{\sf a}}}}\\ \end{array} \left.\vphantom{\begin{array}{c} {{\sf l}^{''}_{{{\sf i}{\sf a}}}}\\ {{\sf m}^{''}_{{{\sf i}{\sf a}}}}\\ \end{array}}\right) \sf Rot \gamma_{{\sf i}{\sf a}} \left(\vphantom{\begin{array}{c} {{\sf l}^{'}_{{\sf i}}}- {{{\sf... ...{'}_{{\sf i}}}- {{{\sf m}^{'}_{{\sf i}}}_{{\sf a}{0}}}\\ \end{array}}\right. \begin{array}{c} {{\sf l}^{'}_{{\sf i}}}- {{{\sf l}^{'}_{{\sf i}... ...{\sf m}^{'}_{{\sf i}}}- {{{\sf m}^{'}_{{\sf i}}}_{{\sf a}{0}}}\\ \end{array} \left.\vphantom{\begin{array}{c} {{\sf l}^{'}_{{\sf i}}}- {{{\sf... ...{'}_{{\sf i}}}- {{{\sf m}^{'}_{{\sf i}}}_{{\sf a}{0}}}\\ \end{array}}\right)$$ (55)

The matrix $\sf Rot$($\gamma_{{\sf i}{\sf a}}$) represents a rotation over the angle $\gamma_{{\sf i}{\sf a}}$ between the positive $\sf l^{'}_{{\sf i}}$-axis of the feed-frame and the $\sf l^{''}_{{{\sf i}{\sf a}}}$-axis of the relevant receptor-frame. For receptor $\sf b$:

$$\left(\vphantom{\begin{array}{c} {{\sf l}^{''}_{{{\sf i}{\sf b}}}}\\ {{\sf m}^{''}_{{{\sf i}{\sf b}}}}\\ \end{array}}\right. \begin{array}{c} {{\sf l}^{''}_{{{\sf i}{\sf b}}}}\\ {{\sf m}^{''}_{{{\sf i}{\sf b}}}}\\ \end{array} \left.\vphantom{\begin{array}{c} {{\sf l}^{''}_{{{\sf i}{\sf b}}}}\\ {{\sf m}^{''}_{{{\sf i}{\sf b}}}}\\ \end{array}}\right) \sf Rot \gamma_{{\sf i}{\sf b}} \left(\vphantom{\begin{array}{c} {{\sf l}^{'}_{{\sf i}}}- {{{\sf... ...{'}_{{\sf i}}}- {{{\sf m}^{'}_{{\sf i}}}_{{\sf b}{0}}}\\ \end{array}}\right. \begin{array}{c} {{\sf l}^{'}_{{\sf i}}}- {{{\sf l}^{'}_{{\sf i}... ...{\sf m}^{'}_{{\sf i}}}- {{{\sf m}^{'}_{{\sf i}}}_{{\sf b}{0}}}\\ \end{array} \left.\vphantom{\begin{array}{c} {{\sf l}^{'}_{{\sf i}}}- {{{\sf... ...{'}_{{\sf i}}}- {{{\sf m}^{'}_{{\sf i}}}_{{\sf b}{0}}}\\ \end{array}}\right)$$ (56)

($\sf l^{'}_{{\sf i}}_{{\sf a}{0}}$,$\sf m^{'}_{{\sf i}}_{{\sf a}{0}}$) and ($\sf l^{'}_{{\sf i}}_{{\sf b}{0}}$,$\sf m^{'}_{{\sf i}}_{{\sf b}{0}}$) represent pointing offsets of receptor $\sf a$ and $\sf b$ respectively. These can be used to model `beam-squint' of feeds that are not axially symmetric.

Matrices and vectors

The following matrices and vectors play a role in the Measurement Equation:

  

 $\vec{I}\,$
\vvIQUV 		 Stokes vector of the source (I,Q,U,V). 

 
$\vec{V}\,$,$\sf v$
\vvCoh,\vvCohEl 		 Coherency vector, and one of its elements. 

 

 $\sf S$
\mmStokes 		 Stokes matrix, conversion between polarisation representations. 

 
$\sf S^{+}_{}$
\mmStokes\ssLin 		 Conversion to linear representation. 

 
$\sf S^{\odot}_{}$
\mmStokes\ssCir 		 Conversion to circular representation. 

 

 $\cal {M}$
\mmMueller 		 Mueller matrix: Stokes to Stokes through optical `element' 

 

 
$\sf X$,$\sf x$
\mmXifr,\mmXifrEl 		 Correlator matrix (4 x 4). 

 
$\sf M$,$\sf m$
\mmMifr,\mmMifrEl 		 Multiplicative interferometer-based gain matrix (4 x 4). 

 
$\vec{A}\,$,$\sf a$
\vvAifr,\vvAifrEl 		 Additive  interferometer-based gain vector.

The following feed-based Jones matrices (2 x 2) have a well-defined meaning:

  

 
$\sf J$,$\sf j$
\mjJones,\mjJonesEl 		 Jones matrix, and one of its elements. 

 

 
$\sf F$,$\sf f$
\mjFrot,\mjFrotEl 		 Faraday rotation (of the plane of linear pol.) 

 
$\sf T$,$\sf t$
\mjTrop,\mjTropEl 		 Atmospheric gain (refraction, extinction). 

 
$\sf P$,$\sf p$
\mjProj,\mjProjEl 		 Projected receptor angle(s) w.r.t. 
$\sf x$,$\sf y$
frame 

 
$\sf B$,$\sf b$
\mjBtot,\mjBtotEl 		 Total feed voltage pattern        (i.e. 
$\sf B$  =  $\sf D$ $\sf E$ $\sf P$. 

 
$\sf E$,$\sf e$
\mjBeam,\mjBeamEl 		 Traditional feed voltage beam. 

 
$\sf C$,$\sf c$
\mjConf,\mjConfEl 		 Feed configuration matrix (...). 

 
$\sf D$,$\sf d$
\mjDrcp,\mjDrcpEl 		 Leakage between receptors $\sf a$
and $\sf b$. 

 
$\sf H$,$\sf h$
\mjHybr,\mjHybrEl 		 Hybrid network, to convert to circular pol. 

 
$\sf G$,$\sf g$
\mjGrec,\mjGrecEl 		 feed-based electronic gain. 

 
$\sf K$,$\sf k$
\mjKern,\mjKernEl 		 Fourier Transform Kernel (baseline phase weight) 

 
$\sf K^{0}_{}$,$\sf k^{0}_{}$
\mjKref,\mjKrefEl 		 FT kernel for the fringe-stopping centre. 

 
$\sf K^{'}_{}$,$\sf k^{'}_{}$
\mjKoff,\mjKoffEl 		 FT kernel relative to the fringe-stopping centre. 

 
$\sf Q$,$\sf q$
\mjQsum,\mjQsumEl 		 Electronic gain of tied-array feed after summing. 

 


Some special matrices and vectors: 

 
$\sf Zero$
\mmZero 		 Zero matrix  

 $\vec{0}\,$
\vvZero 		 Zero vector  

 $\cal {U}$
\mmUnit 		 Unit matrix 

 

 
$\sf Diag$(a, b) \mjDiag 		 Diagonal matrix with elements a, b 

 
$\sf Mult$(a) \mjMult 		 Multiplication with factor a 

 
$\sf Rot$($\alpha$[,$\beta$]) \mjRot 		 [pseudo] Rotation over an angle $\alpha$, $\beta$ 

 
$\sf Ell$($\alpha$[,$\beta$]) \mjEll 		 Ellipticity angle[s] $\alpha$, $\beta$ 

 $\cal {H}$
\mjLtoC 		 Signal conversion from linear to circular. 

 
$\cal {H}$-1 \mjCtoL 		 Signal conversion from circular to linear.

Definitions of some special matrices:

$$\sf Diag \equiv \left(\vphantom{\begin{array}{cc}a & 0\\ 0 & b \end{array}}\right. \begin{array}{cc}a & 0\\ 0 & b \end{array} \left.\vphantom{\begin{array}{cc}a & 0\\ 0 & b \end{array}}\right) \sf Diag \sf Mult \left(\vphantom{\begin{array}{cc}1 & 0\\ 0 & 1 \end{array}}\right. \begin{array}{cc}1 & 0\\ 0 & 1 \end{array} \left.\vphantom{\begin{array}{cc}1 & 0\\ 0 & 1 \end{array}}\right)$$ (57)

A `pure' rotation $\sf Rot$($\alpha$) is a special case of a `pseudo rotation' $\sf Rot$($\alpha$,$\beta$):

$$\sf Rot \alpha \beta \equiv \left(\vphantom{\begin{array}{cc}\cos\alpha & -\sin\alpha\\ \sin\beta & \cos\beta \end{array}}\right. \begin{array}{cc}\cos\alpha & -\sin\alpha\\ \sin\beta & \cos\beta \end{array} \left.\vphantom{\begin{array}{cc}\cos\alpha & -\sin\alpha\\ \sin\beta & \cos\beta \end{array}}\right) \sf Rot \alpha \equiv \sf Rot \alpha \alpha \left(\vphantom{\begin{array}{cc}\cos\alpha & -\sin\alpha\\ \sin\alpha & \cos\alpha \end{array}}\right. \begin{array}{cc}\cos\alpha & -\sin\alpha\\ \sin\alpha & \cos\alpha \end{array} \left.\vphantom{\begin{array}{cc}\cos\alpha & -\sin\alpha\\ \sin\alpha & \cos\alpha \end{array}}\right)$$ (58)

Ellipticity:

$$\sf Ell \alpha \beta \equiv \left(\vphantom{\begin{array}{cc}\cos\alpha & i\sin\alpha\\ -i\sin\beta & \cos\beta \end{array}}\right. \begin{array}{cc}\cos\alpha & i\sin\alpha\\ -i\sin\beta & \cos\beta \end{array} \left.\vphantom{\begin{array}{cc}\cos\alpha & i\sin\alpha\\ -i\sin\beta & \cos\beta \end{array}}\right) \sf Ell \alpha \equiv \sf Ell \alpha \alpha \left(\vphantom{\begin{array}{cc}\cos\alpha & i\sin\alpha\\ i\sin\alpha & \cos\alpha \end{array}}\right. \begin{array}{cc}\cos\alpha & i\sin\alpha\\ i\sin\alpha & \cos\alpha \end{array} \left.\vphantom{\begin{array}{cc}\cos\alpha & i\sin\alpha\\ i\sin\alpha & \cos\alpha \end{array}}\right)$$ (59)

Miscellaneous parameters

  

 $\beta$
\ppParall 		 Parallactic angle, form North pole to zenith 

 HA \ppHA 		 Hour Angle 

 RA \ppRA 		 Right Ascension  

 DEC \ppDEC 		 Declination  

 LAT \ppLAT 		 Latitude on Earth 

 

 $\sf t$
\ccT 		 Time 

 $\sf f$
\ccF 		 Frequency 

 

 $\chi$
\ppFarad 		 Faraday rotation angle

 

 a \ppAmpl 		 Amplitude 

 $\psi$
\ppPhase 		 Phase 

 $\zeta$
\ppPhaseZero 		 Phase zero 

 $\phi$
\ppRcpPosDev 		 Dipole position angle error 

 $\theta$
\ppRcpEllDev 		 receptor ellipticity